# Consider the graph to the right. Explain the idea of a critical value. Then determine which x-values are critical values, and state why. A coordinate plane has a horizontal x-axis and a vertical y-axis. From left to right, ten points on the x-axis are l

Can you help me understand this Calculus question?

Consider the graph to the right. Explain the idea of a critical value. Then determine whichx-values are critical values, and state why. |
A coordinate plane has a horizontal x-axis and a vertical y-axis. From left to right, ten points on the x-axis are labeled from x 1 to x 10. The graph of a function is a continuous curve in quadrant one, and its behavior from left to right is as follows. It falls to the bottom of a U shape at x 1. Then it rises, passing through a point at x 2, to the top of an inverted U shape at x 3. Then it falls to a cusp at x 4 that is higher than the point at x 1. Then it rises to the top of an inverted U shape at x 5 that is lower than the point at x 3. Then it falls to the bottom of a U shape at x 6 that is higher than the point at x 1 but lower than the point at x 4. Then it rises to a cusp at x 7 that is lower than the point at x 4. Then it falls to the bottom of a U shape at x 8 that is lower than the point at x 1. Then it rises, passing through a point at x 9, to the top of an inverted U shape at x 10 that is higher than the point at x 3. Then it falls. |

What is the idea of a critical value?

Critical values are x-values where relative extrema may occur.

Critical values are y-values where relative extrema may occur.

Critical values are the y-values of relative extrema.

Critical values are x-values where relative extrema occur.

Which x-values are critical values?

x 1x1,

x 3x3,

x 5x5,

x 6x6,

x 8x8,

and

x 10x10

because relative extrema occur at these points.

x 4x4

and

x 7x7

because a corner exists at these x-values.

x 1x1,

x 3x3,

x 4x4,

x 5x5,

x 6x6,

x 7x7,

x 8x8,

and

x 10x10

because the slopes of the tangent lines at these points are 0 or do not occur.

All x-values are critical values because they all occur at relative extrema or where the slopes of the tangent lines do not exist.

better explanation in screenshot 🙂